I will present a recent joint work with Andrew Lawrie from MIT. We consider the harmonic map heat flow for maps from the plane $R^2$ to the sphere $S^2$, under the so-called equivariant symmetry. It is known that solutions to the initial value problem exhibit bubbling along a sequence of times - the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by our recent work on the soliton resolution problem for equivariant wave maps.

Numerical semigroups are combinatorial objects that lead to deep and subtle questions.  With tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and computer-aided design, we answer virtually all asymptotic questions about factorization lengths in numerical semigroups.  Our results yield uncannily accurate predictions, along with unexpected results about symmetric functions, trace polynomials, and the statistical properties of certain AF C$^*$-algebras. 

Partially supported by NSF Grants DMS-1800123 and DMS-2054002.  Joint work (in various combinations) with K.~Aguilar, A.~B\"ottcher, \'A. Ch\'avez, L.~Fukshansky, M.~Omar, C.~O'Neill, J.~Vol\v{c}i\v{c} and undergraduate students J.~Hurley, G.~Udell, T.~Wesley, S.~Yih.

 Modified Newton methods are designed to extend the desirable properties of classical Newton method to a wider class of optimization problems. If the Hessian of the objective function is singular at the solution, these methods tend to behave like gradient descent and rapid local convergence is lost. An adaptive regularization technique is described that yields a modified Newton method that retains superlinear local convergence on non-convex problems without the nonsingularity assumption at the solution. The minimum norm perturbation and symmetric indefinite factorization used to construct a sufficiently positive definite approximate Hessian are discussed, and numerical results comparing regularized and standard modified Newton methods will be presented. Lastly, a well-behaved pathological example will be used to illustrate an assumption required for superlinear convergence.

Schubert polynomials originated in the study of the cohomology ring for the complete flag manifold by Bernstein, Gelfand, and Gelfand and Demazure, with their combinatorics, developed extensively by Lascoux and Schutzenberger. For each permutation, there is a Schubert polynomial which, when evaluated at certain Chern classes, gives the cohomology class of a Schubert subvariety of the flag manifold. Thus the Schubert structure constants enumerate flags in a suitable triple intersection of Schubert varieties. As such, they are known from geometry to be nonnegative. A fundamental open problem in algebraic combinatorics is to give a positive combinatorial formula for these structure constants. 

Recently, I conjectured a formula for Schubert structure constants in the classical flag manifold that occurs in the product of an arbitrary Schubert class by one pulled back from a Grassmannian projection. In this talk, I’ll present joint work with Nantel Bergeron in which we define an insertion algorithm on Kohnert diagrams, proving this conjecture. 

This talk should be generally accessible, with no prior knowledge of Schubert varieties or Schubert polynomials required.

The Segal conjecture suggests that the 2-completed $C_2$-equivariant sphere is equivalent to its homotopy completion. However, the genuine $C_2$-equivariant Adams spectral sequence for the $C_2$-equivariant sphere is not isomorphic to the Borel one. In this talk, I will show that the Borel $C_2$-equivariant Adams spectral sequence can be obtained from the genuine one through a degree shifting of the negative cone with connecting differentials shortened. I will also show that the Borel $C_2$-equivariant Adams spectral sequence is related to some classical Adams spectral sequences, whose $E_2$-terms are computable through the Curtis algorithm.

In 1903, Max Dehn settled the following question: which rectangles can you tile with finitely many square tiles (possibly of different sizes)? In this talk, we'll see two (relatively) modern proofs of his result. The first involves redefining the area of a rectangle in a way that would make a measure theorist's skin crawl, and the second involves something even more sacrilegious: physics.

Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis' thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underlying this linkage allows transferring mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behavior of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem; predictably, the latter fails to improve upon the currently best-known bound, achieved via finer methods by Selberg more than half a century ago.

Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraints like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. The study of spaces of metrics and their moduli has been a topic of interest for differential geometers, global and geometric analysts, and topologists alike, and I will introduce to and survey in detail the main results and open questions in the field with a focus on non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics

$p$-adic representations are important objects in number theory, and stable lattices serve as a connection between the study of ordinary and modular representations. These stable lattices can be understood as stable vertices in Bruhat-Tits buildings. From this viewpoint, the study of fixed point sets in these buildings can aid research on $p$-adic representations. The simplicial distance holds an important role as it connects the combinatorics of lattices and the geometry of root systems. In particular, the fixed-point sets of Moy-Prasad subgroups are precisely the simplicial balls. In this talk, I'll explain those findings and compute their simplicial volume under certain conditions.

[pre-talk at 1:20PM]